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Cristina Giannotti

Publications and source records attributed to Cristina Giannotti.

11 recordsLinked to original sources
arXiv2026

Null fluid gravitational fields on Kerr manifolds and optical lifts of Sasaki structures

Building on the characterisation in [C. D. Hill, J. Lewandowski and P. Nurowski, Indiana Univ. Math. J. 57 (2008), 3131--3176] of 4-dimensional Lorentzian metrics adapted to an optical structure and satisfying the null fluid Einstein equations, we give an explicit parameterisation of this class under the assumption that the optical structure is of Kerr type. As immediate consequences, we obtain: (1) a new method for constructing solutions to the Einstein equations with a null fluid energy momentum tensor, yielding a large family of explicit metrics that naturally includes the classical Kerr black hole metrics and all Ricci flat examples described in [M. Ganji, C. Giannotti, G. Schmalz and A. Spiro, Ann. Physics 75 (2025), Paper No. 169908, 28]; (2) a solution to a conjecture in Hill, Lewandowski and Nurowski's paper on the local existence of smooth optical lifts in the case of Sasaki CR structures.

gr-qc↗
arXiv2026

Black holes and black regions, horizons and barriers in Lorentzian manifolds

We prove that if S is a time-oriented null hypersurface of a Lorentzian n-manifold (M, g), the causal world-lines, which intersect transversally S and are time-oriented in a compatible way, cross the hypersurface all in the same direction, the other being forbidden. Even if it is known that a smooth event horizon (in the sense of Penrose, Hawking and Ellis) is a null hypersurface and has the above semi-permeability property, at the best of our knowledge, in the literature it was not stated so far that the latter is a mere consequence of the former. Our result leads to the concepts of barriers (= null hypersurfaces separating the space-time into disjoint regions) and black regions (= time-oriented regions bounded by barriers). These objects naturally include (smooth) event horizons and (smoothly bounded) black holes. Since barriers are defined by two simple properties -- the merely local property of "nullity" combined with the global property of "separating the space-time" -- we expect they may be used to simplify computations for locating static and/or dynamic horizons in numerical computations.

gr-qc↗
arXiv2025

Flows of vector fields and the Kalman Theorem

We give two proofs of the Kalman Theorem, alternative to the most common ones, which infer such a classical result of Control Theory using just very basic facts on flows of vector fields. These proofs are apt to be generalised in diverse directions -- in fact one of them has been already generalised, yielding new criteria for local controllability of non-linear real analytic controlled systems.

math.OC↗
arXiv2024

Einstein manifolds with optical geometries of Kerr type

We classify the Ricci flat Lorentzian $n$-manifolds satisfying three particular conditions, encoding and combining some crucial features of the Kerr metrics and the Robinson-Trautman optical structures. We prove that: (a) If $n>4$, there is no Lorentzian manifold satisfying the considered Kerr type conditions, in unexpected contrast with what occurs for the metrics satisfying (very similar) Taub-NUT type conditions; (b) If $n=4$ there are two large classes of such Kerr type manifolds. Each class consists of manifolds fibering over open Riemann surfaces, equipped with a metric of constant Gaussian curvature $κ= 1$ or $κ= -1$. The first class includes a three parameter family of metrics admitting real analytic extensions to $(\mathbb R^3 \setminus\{0\}) \times \mathbb R = (S^2 \times \mathbb R_+) \times \mathbb R$ and a large class of other metrics not admitting this kind of extensions. The metrics of this first class admitting such extensions are all isometric to the well known Kerr metrics, with the three parameters corresponding to the three space-like components of the angular momentum of the gravitational field. The second class contains a subclass of metrics defined on $\big(\mathbb D\times \mathbb R_+\big)\times \mathbb R$, where $\mathbb D$ is the Lobachevsky Poincaré disc. This subclass is in bijection with the holomorphic functions on $\mathbb D$ satisfying an appropriate open condition. These and other results are consequences of a very simple way to construct totally explicit examples of Ricci flat Lorentzian manifolds.

math.DG↗
arXiv2024

Proving the Chow-Rashevskii Theorem à la Rashevskii

We give a new independent proof of a generalised version of the theorem by Rashevskii, which appeared in [Uch. Zapiski Ped. Inst. K. 2 (1938), 83 -- 94] and from which the classical Chow-Rashevskii Theorem follows as a corollary. The proof is structured to allow generalisations to the case of orbits of compositions of flows in absence of group structures, thus appropriate for applications in Control Theory. In fact, the same structure of the proof has been successfully exploited in [C. Giannotti, A. Spiro and M. Zoppello, arXiv 2401.07555 \& 2401.07560 (2024)] to determine new controllability criteria for real analytic non-linear control systems. It also yields a corollary, which can be used to derive results under lower regularity assumptions, as it is illustrated by a simple explicit example.

math.DG↗
arXiv2024

Distributions and controllability problems (I)

We consider a non-linear real analytic control system of first order $\dot q^i = f^i(t, q, w)$, with controls $w = (w^α)$ in a connected open set $\mathcal{K} \subset \mathbb{R}^m$ and configurations $q = (q^i)$ in $\mathcal{Q} := \mathbb{R}^n$. The set of points in the extended space-time $\mathcal{M} = \mathbb{R} \times \mathcal{Q} \times \mathcal{K}$, which can be reached from a triple $x_o = (t_o , q_o, w_o) \in \mathcal{M}$ through a continuous graph completion $γ(s) = \big(t(s), q(t(s)), w(t(s))\big)$ of the graph of a solution $t \to (q(t), w(t))$, $t \in [t_o ,t_o + T]$, with piecewise real analytic controls, is called the {\it $\mathcal{M}$-attainable set of $x_o$ in time $T$}. We prove that if $y_o$ is an $\mathcal{M}$-attainable point of $x_o$, a large set of other nearby $\mathcal{M}$-attainable points of $x_o$ can be determined starting directly from $y_o$ and applying an appropriate ordered composition of flows of vector fields in a distinguished distribution $\mathcal{D}^{II} \subset T \mathcal{M}$, canonically associated with the control system. We then determine sufficient conditions for such neighbouring points to constitute an orbit of the pseudogroup of local diffeomorphisms generated by the vector fields in $\mathcal{D}^{II}$. If such conditions are satisfied and if the tangent spaces of these orbits have maximal rank projections onto $\mathcal{Q}$, the control system is locally accessible and has the small time local controllability property near the state points of equilibrium. These results lead to new proofs of classical local controllability criterions and yield new methods to establish the accessibility and the small time local controllability of non-linear control systems.

math.OC↗
arXiv2024

Distributions and controllability problems (II)

In [C. Giannotti, A. Spiro, M. Zoppello, {\it Distributions and controllability problems (I)}, preprint posted on ArXiv (2024)], we introduced a new approach to the real analytic non-linear control systems of the form $\dot q^i = f^i(t, q, w)$, with controls $w = (w^α)$ running in a connected open set $\mathcal{K}$ of $ \mathbb{R}^m$ and states represented by points $q = (q^i)$ in a configuration space $\mathcal{Q} := \mathbb{R}^n$. The new approach consists of a differential-geometric study of (a) the oriented piecewise regular curves in the {\it extended space-time} $\mathcal{M} = \mathbb{R} \times \mathcal{Q} \times \mathcal{K}$, which are the (completed) graphs of the piecewise real analytic solutions $t \mapsto (q(t), w(t))$ of the control system, and (b) the local structure of the sets of points of $\mathcal{M}$ that are reachable from an initial point $x_o = (t_o, q_o, w_o) \in \mathcal{M}$ through such (completed) graphs. The main results of that paper are two new criterions which can be used to establish the small time local controllability near stable points of real analytic non-linear systems. The goal of this paper is to offer a friendly user's guide to those criterions, illustrating them by several examples. In particular, we analyse certain non-linear control systems, for which the new criterions show that they are small time locally controllable at their stable points, while, at the best of our knowledge, all other previous criterions are either inconclusive or not applicable.

math.OC↗
arXiv2023

On the Pontryagin Maximum Principle under differential constraints of higher order

Exploiting our previous results on higher order controlled Lagrangians in [Nonlinear Anal. {\bf 207} (2021), 112263], we derive here an analogue of the classical first order Pontryagin Maximum Principle (PMP) for cost minimising problems subjected to higher order differential constraints $\frac{d^k x^j}{dt^k} = f^j\big(t, x(t), \frac{d x}{dt}(t), \ldots, \frac{d^{k-1} x}{dt^{k-1}}(t), u(t)\big)$, $t \in [0,T]$, where $u(t)$ is a control curve in a compact set $K \subset \mathbb R^m$. This result and its proof can be considered as a detailed illustration of one of the claims of that previous paper, namely that the results of that paper, originally established in a smooth differential geometric framework, yield directly properties holding under much weaker and more common assumptions. In addition, for further clarifying our motivations, in the last section we display a couple of quick indications on how the two-step approach of this paper (i.e., a preliminary easy-to-get differential geometric discussion followed by a refining analysis to weaken the regularity assumptions) might be fruitfully exploited also in the context of control problems governed by partial differential equations or in studies on the dynamics of controlled mechanical systems.

math.OC↗
arXiv2021

Control problems with differential constraints of higher order

We consider cost minimising control problems, in which the dynamical system is constrained by higher order differential equations of Euler-Lagrange type. Following ideas from a previous paper by the first and the third author, we prove that a curve of controls $u_o(t)$ and a set of initial conditions $σ_o$ gives an optimal solution for a control problem of the considered type if and only if an appropriate double integral is greater than or equal to zero along any homotopy $(u(t, s), σ(s))$ of control curves and initial data starting from $u_o(t) = u(t, 0)$ and $σ_o = σ(0)$. This property is called "Principle of Minimal Labour". From this principle we derive a generalisation of the classical Pontryagin Maximum Principle that holds under higher order differential constraints of Euler-Lagrange type and without the hypothesis of fixed initial data.

math.OC↗
arXiv2010

Steepest descent curves of convex functions on surfaces of constant curvature

Let S be a complete surface of constant curvature K = + 1 or -1, i.e. the sphere S^2 or the Lobachevskij plane L^2, and D a bounded convex subset of S. If S = S^2, assume also diameter (D) < pi/2. It is proved that the length of any steepest descent curve of a quasi-convex function in D is less than or equal to the perimeter of D. This upper bound is actually proved for the class of G-curves, a family of curves that naturally includes all steepest descent curves. In case S = S^2, it is also proved the existence of G-curves, whose length is equal to the perimeter of their convex hull, showing that the above estimate is indeed optimal. The results generalize theorems by Manselli and Pucci on steepest descent curves in the Euclidean plane.

math.CA↗